Andreev billiards

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This is a review of recent advances in our understanding of how Andreev reflection at a superconductor modifies the excitation spectrum of a quantum dot. The emphasis is on two-dimensional impurity-free structures in which the classical dynamics is chaotic. Such Andreev billiards differ in a fundamental way from their non-superconducting counterparts. Most notably, the difference between chaotic and integrable classical dynamics shows up already in the level density, instead of only in the level--level correlations. A chaotic billiard has a gap in the spectrum around the Fermi energy, while integrable billiards have a linearly vanishing density of states. The excitation gap Egap corresponds to a time scale h/Egap which is classical (h-independent, equal to the mean time tdwell between Andreev reflections) if tdwell is sufficiently large. There is a competing quantum time scale, the Ehrenfest time tₑ, which depends logarithmically on h. Two phenomenological theories provide a consistent description of the tₑ-dependence of the gap, given qualitatively by Egap min(h/tdwell,h/tₑ). The analytical predictions have been tested by computer simulations but not yet experimentally.